One major idea in mathematics is the idea of “closure”. This is the ques-tion: What is the set of all things that can result from my proposed oper-ations? In the case of vectors: What is the set of vectors that can result bystarting with a small set of vectors, and adding them to each other andscaling them? This results in a vector space

In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."

If you do the math

four

pi = 3.1415 ... e = 2.718 ... Euler's constant, gamma = 0.577215 ... = lim n -> infinity > (1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n)) (Not proven to be transcendental, but generally believed to be by mathematicians.) Catalan's constant, G = sum (-1)^k / (2k + 1 )^2 = 1 - 1/9 + 1/25 - 1/49 + ... (Not proven to be transcendental, but generally believed to be by mathematicians.) Liouville's number 0.110001000000000000000001000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere. Chaitin's "constant", the probability that a random algorithm halts. (Noam Elkies of Harvard notes that not only is this number transcendental but it is also incomputable.) Chapernowne's number, 0.12345678910111213141516171819202122232425... This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?) Special values of the zeta function, such as zeta (3). (Transcendental functions can usually be expected to give transcendental results at rational points.) ln(2). Hilbert's number, 2(sqrt 2 ). (This is called Hilbert's number because the proof of whether or not it is transcendental was one of Hilbert's famous problems. In fact, according to the Gelfond-Schneider theorem, any number of the form ab is transcendental where a and b are algebraic (a ne 0, a ne 1 ) and b is not a rational number. Many trigonometric or hyperbolic functions of non-zero algebraic numbers are transcendental.) epi pie (Not proven to be transcendental, but generally believed to be by mathematicians.) Morse-Thue's number, 0.01101001 ... ii = 0.207879576... (Here i is the imaginary number sqrt(-1). Isn't this a real beauty? How many people have actually considered rasing i to the i power? If a is algebraic and b is algebraic but irrational then ab is transcendental. Since i is algebraic but irrational, the theorem applies. Note also: ii is equal to e(- pi / 2 ) and several other values. Consider ii = e(i log i ) = e( i times i pi / 2 ) . Since log is multivalued, there are other possible values for ii. Here is how you can compute the value of ii = 0.207879576... 1. Since e^(ix) = Cos x + i Sin x, then let x = Pi/2. 2. Then e^(iPi/2) = i = Cos Pi/2 + i Sin Pi/2; since Cos Pi/2 = Cos 90 deg. = 0. But Sin 90 = 1 and i Sin 90 deg. = (i)*(1) = i. 3. Therefore e^(iPi/2) = i. 4. Take the ith power of both sides, the right side being i^i and the left side = [e^(iPi/2)]^i = e^(-Pi/2). 5. Therefore i^i = e^(-Pi/2) = .207879576... Feigenbaum numbers, e.g. 4.669 ... . (These are related to properties of dynamical systems with period-doubling. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669 ... , and it has been discovered in many physical systems before they enter the chaotic regime. It has not been proven to be transcendental, but is generally believed to be.)

A rooted binary tree is full if every vertex has either two children or no children.

computer owned by Patrick Laroche of Ocala, Fla., discovered the number on Dec. 7,

Basic Statistics

Mathematics for Machine Learning

Начните с нескольких основных книг, которые изменят ваши представления о математикеЛучше всего «ставит голову» книга «Начала теории множеств» Николая Верещагина и Александра Шеня. Она даст основу для понимания (а в дальнейшем и самостоятельного построения!) логических рассуждений. С нее же начнется и понимание теории множеств, лежащей в основе современной математики.Охватить больше разделов математики поможет книга Рихарда Куранта и Герберта Роббинса «Что такое математика?». Как и книгу Верещагина и Шеня, эту книгу нужно читать внимательно, делая все упражнения. Если первые две книги окажутся сложными, можете начать с моей «Математики для гуманитариев». Ее также следует читать с самого начала, страницу за страницей, не стоит браться за чтение с середины. Она не очень простая, но предварительных сведений и математической культуры не предполагает.Чтобы понять, каким образом математика входит в нашу жизнь, можно прочитать «Кому нужна математика?» Андрея Райгородского и Нелли Литвак или «Математическую составляющую» (сборник сюжетов под редакцией Николая Андреева).

Learning Math Earlier With Computers

we found advanced math and science courses advertised but not actually offered and specialized STEM programs eroded or discontinued within a few years of inception.

Yet, across the country, 2 in 5 high schools don't offer physics, according to an Education Week Research Center analysis of data from the U.S. Department of Education's office for civil rights.

n this note we determine thetorsion subgroup of the additive group of (1.1) in the case whereLhas rank 2, thatisXis a set of two elements, andcis a prime number

Of particular interest are the lower central series: G1=GG1=GG_1 =G, Gi+1=[Gi,G]Gi+1=[Gi,G]G_{i+1} = [G_i, G], i≥1i≥1i \ge 1, and, for a fixed prime number p, the Zassenhaus series (see [7, 8]).

Leonhard Euler, who adopted it in 1737

Let’s use some common units as examples: gram (g), erg (erg), and solar mass per cubic megaparsec (Msun / Mpc33^3). g is an atomic, CGS base unit, erg is an atomic unit in CGS, but is not a base unit, and Msun/Mpc33^3 is a combination of atomic units, which are not in CGS, and one of them even has an SI prefix. The dimensions of g are mass and the cgs factor is 1. The dimensions of erg are mass * length$^2$ * time−2−2^{-2} and the cgs factor is 1. The dimensions of Msun/Mpc33^3 are mass / length33^3 and the cgs factor is about 6.8e-41.

satisfy the two usual distributive laws. Singlesemirings as well as classes of semirings form important structures in Automataand Formal Languages Theories [5

1st order Eulerian numbers:Permutations

The first review, by C. Hendricks Brown et al., poses the issues raised by the growingrecognition

For detailed study of various properties, generalization and application of Wrightfunction and generalized Wright function,

Whether you're a student, parent, or teacher, this book is your key to unlocking the aha! moments that make math click -- and learning enjoyable.

LSB (least significant bit):  The rightmost bit MSB (most significant bit):  The leftmost bit

x2

The first set, called Math Instructional, was for apps that would make math relevant for students by linking it to their lives and enabling students at different ability levels to work together

P(B|E) = P(B) X P(E|B) / P(E), with P standing for probability, B for belief and E for evidence. P(B) is the probability that B is true, and P(E) is the probability that E is true. P(B|E) means the probability of B if E is true, and P(E|B) is the probability of E if B is true.

The probability that a belief is true given new evidence equals the probability that the belief is true regardless of that evidence times the probability that the evidence is true given that the belief is true divided by the probability that the evidence is true regardless of whether the belief is true. Got that?

θ dμ ≥ p 16 π | Σ |

GIBBONS-PENROSE INEQUALITY

θ dμ ≥ p 16 π | Σ |

A function like f(x,y)=x+y is a function of two variables. It takes an element of R2, like (2,1), and gives a value that is a real number (i.e., an element of R), like f(2,1)=3. Since f maps R2 to R, we write f:R2→R. We can also use this “mapping” notation to define the actual function. We could define the above f(x,y) by writing f:(x,y)↦x+y. To contrast a simple real number with a vector, we refer to the real number as a scalar. Hence, we can refer to f:R2→R as a scalar-valued function of two variables or even just say it is a real-valued function of two variables. Everything works the same for scalar valued functions of three or more variables. For example, f(x,y,z), which we can write f:R3→R, is a scalar-valued function of three variables.

f:R→R as a shorthand way of expressing that f is a function from R onto R.

A computable Dedekind cut is a computable function which when provided with a rational number as input returns or ,